Use mathematical induction to prove that the statement is true for every positive integer n.0.35n < 1

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 0.351 =0.35 < 1. So, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 0.35k < 1
 Multiply both sides by 0.35:
 0.35 ? 0.35k = 0.35k + 1 < 0.35 < 1 or 0.35k + 1 < 1
Since the statement is true for n = k + 1 as long as it is true for n = k, and since the statement is true for n = 1, then it is true for all natural numbers n.